Theorem pnt 22747

Description: The Prime Number Theorem: the number of prime numbers less than x tends asymptotically to x / log ( x ) as x goes to infinity. (Contributed by Mario Carneiro, 1-Jun-2016.)

Assertion

Ref	Expression
pnt	|- ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) ~~> r 1

Proof of Theorem pnt

Dummy variables w y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1re 9372	. . . . . . 7 |- 1 e. RR
2	1	rexri 9423	. . . . . 6 |- 1 e. RR*
3		1lt2 10475	. . . . . 6 |- 1 < 2
4		df-ioo 11291	. . . . . . 7 |- (,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x < z /\ z < y ) } )
5		df-ico 11293	. . . . . . 7 |- [,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x <_ z /\ z < y ) } )
6		xrltletr 11118	. . . . . . 7 |- ( ( 1 e. RR* /\ 2 e. RR* /\ w e. RR* ) -> ( ( 1 < 2 /\ 2 <_ w ) -> 1 < w ) )
7	4, 5, 6	ixxss1 11305	. . . . . 6 |- ( ( 1 e. RR* /\ 1 < 2 ) -> ( 2 [,) +oo ) C_ ( 1 (,) +oo ) )
8	2, 3, 7	mp2an 665	. . . . 5 |- ( 2 [,) +oo ) C_ ( 1 (,) +oo )
9		resmpt 5144	. . . . 5 |- ( ( 2 [,) +oo ) C_ ( 1 (,) +oo ) -> ( ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) |` ( 2 [,) +oo ) ) = ( x e. ( 2 [,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) )
10	8, 9	mp1i 12	. . . 4 |- ( T. -> ( ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) |` ( 2 [,) +oo ) ) = ( x e. ( 2 [,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) )
11	8	sseli 3340	. . . . . . . . . 10 |- ( x e. ( 2 [,) +oo ) -> x e. ( 1 (,) +oo ) )
12		ioossre 11344	. . . . . . . . . . 11 |- ( 1 (,) +oo ) C_ RR
13	12	sseli 3340	. . . . . . . . . 10 |- ( x e. ( 1 (,) +oo ) -> x e. RR )
14	11, 13	syl 16	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> x e. RR )
15		2re 10378	. . . . . . . . . . 11 |- 2 e. RR
16		pnfxr 11079	. . . . . . . . . . 11 |- +oo e. RR*
17		elico2 11346	. . . . . . . . . . 11 |- ( ( 2 e. RR /\ +oo e. RR* ) -> ( x e. ( 2 [,) +oo ) <-> ( x e. RR /\ 2 <_ x /\ x < +oo ) ) )
18	15, 16, 17	mp2an 665	. . . . . . . . . 10 |- ( x e. ( 2 [,) +oo ) <-> ( x e. RR /\ 2 <_ x /\ x < +oo ) )
19	18	simp2bi 997	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> 2 <_ x )
20		chtrpcl 22397	. . . . . . . . 9 |- ( ( x e. RR /\ 2 <_ x ) -> ( theta ` x ) e. RR+ )
21	14, 19, 20	syl2anc 654	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> ( theta ` x ) e. RR+ )
22		0red 9374	. . . . . . . . . . 11 |- ( x e. ( 1 (,) +oo ) -> 0 e. RR )
23	1	a1i 11	. . . . . . . . . . 11 |- ( x e. ( 1 (,) +oo ) -> 1 e. RR )
24		0lt1 9849	. . . . . . . . . . . 12 |- 0 < 1
25	24	a1i 11	. . . . . . . . . . 11 |- ( x e. ( 1 (,) +oo ) -> 0 < 1 )
26		eliooord 11342	. . . . . . . . . . . 12 |- ( x e. ( 1 (,) +oo ) -> ( 1 < x /\ x < +oo ) )
27	26	simpld 456	. . . . . . . . . . 11 |- ( x e. ( 1 (,) +oo ) -> 1 < x )
28	22, 23, 13, 25, 27	lttrd 9519	. . . . . . . . . 10 |- ( x e. ( 1 (,) +oo ) -> 0 < x )
29	13, 28	elrpd 11012	. . . . . . . . 9 |- ( x e. ( 1 (,) +oo ) -> x e. RR+ )
30	11, 29	syl 16	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> x e. RR+ )
31	21, 30	rpdivcld 11031	. . . . . . 7 |- ( x e. ( 2 [,) +oo ) -> ( ( theta ` x ) / x ) e. RR+ )
32	31	adantl 463	. . . . . 6 |- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> ( ( theta ` x ) / x ) e. RR+ )
33		ppinncl 22396	. . . . . . . . . . 11 |- ( ( x e. RR /\ 2 <_ x ) -> (π ` x ) e. NN )
34	14, 19, 33	syl2anc 654	. . . . . . . . . 10 |- ( x e. ( 2 [,) +oo ) -> (π ` x ) e. NN )
35	34	nnrpd 11013	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> (π ` x ) e. RR+ )
36	13, 27	rplogcld 21962	. . . . . . . . . 10 |- ( x e. ( 1 (,) +oo ) -> ( log ` x ) e. RR+ )
37	11, 36	syl 16	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> ( log ` x ) e. RR+ )
38	35, 37	rpmulcld 11030	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> ( (π ` x ) x. ( log ` x ) ) e. RR+ )
39	21, 38	rpdivcld 11031	. . . . . . 7 |- ( x e. ( 2 [,) +oo ) -> ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) e. RR+ )
40	39	adantl 463	. . . . . 6 |- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) e. RR+ )
41	30	ssriv 3348	. . . . . . . 8 |- ( 2 [,) +oo ) C_ RR+
42		resmpt 5144	. . . . . . . 8 |- ( ( 2 [,) +oo ) C_ RR+ -> ( ( x e. RR+ |-> ( ( theta ` x ) / x ) ) |` ( 2 [,) +oo ) ) = ( x e. ( 2 [,) +oo ) |-> ( ( theta ` x ) / x ) ) )
43	41, 42	ax-mp 5	. . . . . . 7 |- ( ( x e. RR+ |-> ( ( theta ` x ) / x ) ) |` ( 2 [,) +oo ) ) = ( x e. ( 2 [,) +oo ) |-> ( ( theta ` x ) / x ) )
44		pnt2 22746	. . . . . . . 8 |- ( x e. RR+ |-> ( ( theta ` x ) / x ) ) ~~> r 1
45		rlimres 13019	. . . . . . . 8 |- ( ( x e. RR+ |-> ( ( theta ` x ) / x ) ) ~~> r 1 -> ( ( x e. RR+ |-> ( ( theta ` x ) / x ) ) |` ( 2 [,) +oo ) ) ~~> r 1 )
46	44, 45	mp1i 12	. . . . . . 7 |- ( T. -> ( ( x e. RR+ |-> ( ( theta ` x ) / x ) ) |` ( 2 [,) +oo ) ) ~~> r 1 )
47	43, 46	syl5eqbrr 4314	. . . . . 6 |- ( T. -> ( x e. ( 2 [,) +oo ) |-> ( ( theta ` x ) / x ) ) ~~> r 1 )
48		chtppilim 22608	. . . . . . 7 |- ( x e. ( 2 [,) +oo ) |-> ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) ) ~~> r 1
49	48	a1i 11	. . . . . 6 |- ( T. -> ( x e. ( 2 [,) +oo ) |-> ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) ) ~~> r 1 )
50		ax-1ne0 9338	. . . . . . 7 |- 1 =/= 0
51	50	a1i 11	. . . . . 6 |- ( T. -> 1 =/= 0 )
52	39	rpne0d 11019	. . . . . . 7 |- ( x e. ( 2 [,) +oo ) -> ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) =/= 0 )
53	52	adantl 463	. . . . . 6 |- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) =/= 0 )
54	32, 40, 47, 49, 51, 53	rlimdiv 13106	. . . . 5 |- ( T. -> ( x e. ( 2 [,) +oo ) |-> ( ( ( theta ` x ) / x ) / ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) ) ) ~~> r ( 1 / 1 ) )
55	14	recnd 9399	. . . . . . . . . 10 |- ( x e. ( 2 [,) +oo ) -> x e. CC )
56		chtcl 22331	. . . . . . . . . . . . 13 |- ( x e. RR -> ( theta ` x ) e. RR )
57	13, 56	syl 16	. . . . . . . . . . . 12 |- ( x e. ( 1 (,) +oo ) -> ( theta ` x ) e. RR )
58	57	recnd 9399	. . . . . . . . . . 11 |- ( x e. ( 1 (,) +oo ) -> ( theta ` x ) e. CC )
59	11, 58	syl 16	. . . . . . . . . 10 |- ( x e. ( 2 [,) +oo ) -> ( theta ` x ) e. CC )
60	55, 59	mulcomd 9394	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> ( x x. ( theta ` x ) ) = ( ( theta ` x ) x. x ) )
61	60	oveq2d 6096	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> ( ( ( theta ` x ) x. ( (π ` x ) x. ( log ` x ) ) ) / ( x x. ( theta ` x ) ) ) = ( ( ( theta ` x ) x. ( (π ` x ) x. ( log ` x ) ) ) / ( ( theta ` x ) x. x ) ) )
62	38	rpcnd 11016	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> ( (π ` x ) x. ( log ` x ) ) e. CC )
63	30	rpne0d 11019	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> x =/= 0 )
64	21	rpne0d 11019	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> ( theta ` x ) =/= 0 )
65	62, 55, 59, 63, 64	divcan5d 10120	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> ( ( ( theta ` x ) x. ( (π ` x ) x. ( log ` x ) ) ) / ( ( theta ` x ) x. x ) ) = ( ( (π ` x ) x. ( log ` x ) ) / x ) )
66	61, 65	eqtrd 2465	. . . . . . 7 |- ( x e. ( 2 [,) +oo ) -> ( ( ( theta ` x ) x. ( (π ` x ) x. ( log ` x ) ) ) / ( x x. ( theta ` x ) ) ) = ( ( (π ` x ) x. ( log ` x ) ) / x ) )
67	38	rpne0d 11019	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> ( (π ` x ) x. ( log ` x ) ) =/= 0 )
68	59, 55, 59, 62, 63, 67, 64	divdivdivd 10141	. . . . . . 7 |- ( x e. ( 2 [,) +oo ) -> ( ( ( theta ` x ) / x ) / ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) ) = ( ( ( theta ` x ) x. ( (π ` x ) x. ( log ` x ) ) ) / ( x x. ( theta ` x ) ) ) )
69	34	nncnd 10325	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> (π ` x ) e. CC )
70	37	rpcnd 11016	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> ( log ` x ) e. CC )
71	37	rpne0d 11019	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> ( log ` x ) =/= 0 )
72	69, 55, 70, 63, 71	divdiv2d 10126	. . . . . . 7 |- ( x e. ( 2 [,) +oo ) -> ( (π ` x ) / ( x / ( log ` x ) ) ) = ( ( (π ` x ) x. ( log ` x ) ) / x ) )
73	66, 68, 72	3eqtr4d 2475	. . . . . 6 |- ( x e. ( 2 [,) +oo ) -> ( ( ( theta ` x ) / x ) / ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) ) = ( (π ` x ) / ( x / ( log ` x ) ) ) )
74	73	mpteq2ia 4362	. . . . 5 |- ( x e. ( 2 [,) +oo ) |-> ( ( ( theta ` x ) / x ) / ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) ) ) = ( x e. ( 2 [,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) )
75		1div1e1 10011	. . . . 5 |- ( 1 / 1 ) = 1
76	54, 74, 75	3brtr3g 4311	. . . 4 |- ( T. -> ( x e. ( 2 [,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) ~~> r 1 )
77	10, 76	eqbrtrd 4300	. . 3 |- ( T. -> ( ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) |` ( 2 [,) +oo ) ) ~~> r 1 )
78		ppicl 22353	. . . . . . . . . 10 |- ( x e. RR -> (π ` x ) e. NN0 )
79	13, 78	syl 16	. . . . . . . . 9 |- ( x e. ( 1 (,) +oo ) -> (π ` x ) e. NN0 )
80	79	nn0red 10624	. . . . . . . 8 |- ( x e. ( 1 (,) +oo ) -> (π ` x ) e. RR )
81	29, 36	rpdivcld 11031	. . . . . . . 8 |- ( x e. ( 1 (,) +oo ) -> ( x / ( log ` x ) ) e. RR+ )
82	80, 81	rerpdivcld 11041	. . . . . . 7 |- ( x e. ( 1 (,) +oo ) -> ( (π ` x ) / ( x / ( log ` x ) ) ) e. RR )
83	82	recnd 9399	. . . . . 6 |- ( x e. ( 1 (,) +oo ) -> ( (π ` x ) / ( x / ( log ` x ) ) ) e. CC )
84	83	adantl 463	. . . . 5 |- ( ( T. /\ x e. ( 1 (,) +oo ) ) -> ( (π ` x ) / ( x / ( log ` x ) ) ) e. CC )
85		eqid 2433	. . . . 5 |- ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) = ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) )
86	84, 85	fmptd 5855	. . . 4 |- ( T. -> ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) : ( 1 (,) +oo ) --> CC )
87	12	a1i 11	. . . 4 |- ( T. -> ( 1 (,) +oo ) C_ RR )
88	15	a1i 11	. . . 4 |- ( T. -> 2 e. RR )
89	86, 87, 88	rlimresb 13026	. . 3 |- ( T. -> ( ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) ~~> r 1 <-> ( ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) |` ( 2 [,) +oo ) ) ~~> r 1 ) )
90	77, 89	mpbird 232	. 2 |- ( T. -> ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) ~~> r 1 )
91	90	trud 1371	1 |- ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) ~~> r 1

Syntax hints:   <-> wb 184   /\ w3a 958   = wceq 1362   T. wtru 1363   e. wcel 1755   =/= wne 2596   C_ wss 3316   class class class wbr 4280   e. cmpt 4338   |` cres 4829   ` cfv 5406  (class class class)co 6080   CCcc 9267   RRcr 9268   0cc0 9269   1c1 9270   x. cmul 9274   +oocpnf 9402   RR*cxr 9404   < clt 9405   <_ cle 9406   / cdiv 9980   NNcn 10309   2c2 10358   NN0cn0 10566   RR+crp 10978   (,)cioo 11287   [,)cico 11289   ~~> r crli 12946   logclog 21890   thetaccht 22312  πcppi 22315

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9325  ax-resscn 9326  ax-1cn 9327  ax-icn 9328  ax-addcl 9329  ax-addrcl 9330  ax-mulcl 9331  ax-mulrcl 9332  ax-mulcom 9333  ax-addass 9334  ax-mulass 9335  ax-distr 9336  ax-i2m1 9337  ax-1ne0 9338  ax-1rid 9339  ax-rnegex 9340  ax-rrecex 9341  ax-cnre 9342  ax-pre-lttri 9343  ax-pre-lttrn 9344  ax-pre-ltadd 9345  ax-pre-mulgt0 9346  ax-pre-sup 9347  ax-addf 9348  ax-mulf 9349

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-iin 4162  df-disj 4251  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309  df-om 6466  df-1st 6566  df-2nd 6567  df-supp 6680  df-recs 6818  df-rdg 6852  df-1o 6908  df-2o 6909  df-oadd 6912  df-er 7089  df-map 7204  df-pm 7205  df-ixp 7252  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-fsupp 7609  df-fi 7649  df-sup 7679  df-oi 7712  df-card 8097  df-cda 8325  df-pnf 9407  df-mnf 9408  df-xr 9409  df-ltxr 9410  df-le 9411  df-sub 9584  df-neg 9585  df-div 9981  df-nn 10310  df-2 10367  df-3 10368  df-4 10369  df-5 10370  df-6 10371  df-7 10372  df-8 10373  df-9 10374  df-10 10375  df-n0 10567  df-z 10634  df-dec 10743  df-uz 10849  df-q 10941  df-rp 10979  df-xneg 11076  df-xadd 11077  df-xmul 11078  df-ioo 11291  df-ioc 11292  df-ico 11293  df-icc 11294  df-fz 11424  df-fzo 11532  df-fl 11625  df-mod 11692  df-seq 11790  df-exp 11849  df-fac 12035  df-bc 12062  df-hash 12087  df-shft 12539  df-cj 12571  df-re 12572  df-im 12573  df-sqr 12707  df-abs 12708  df-limsup 12932  df-clim 12949  df-rlim 12950  df-o1 12951  df-lo1 12952  df-sum 13147  df-ef 13335  df-e 13336  df-sin 13337  df-cos 13338  df-pi 13340  df-dvds 13518  df-gcd 13673  df-prm 13746  df-pc 13886  df-struct 14158  df-ndx 14159  df-slot 14160  df-base 14161  df-sets 14162  df-ress 14163  df-plusg 14233  df-mulr 14234  df-starv 14235  df-sca 14236  df-vsca 14237  df-ip 14238  df-tset 14239  df-ple 14240  df-ds 14242  df-unif 14243  df-hom 14244  df-cco 14245  df-rest 14343  df-topn 14344  df-0g 14362  df-gsum 14363  df-topgen 14364  df-pt 14365  df-prds 14368  df-xrs 14422  df-qtop 14427  df-imas 14428  df-xps 14430  df-mre 14506  df-mrc 14507  df-acs 14509  df-mnd 15397  df-submnd 15447  df-mulg 15527  df-cntz 15814  df-cmn 16258  df-psmet 17652  df-xmet 17653  df-met 17654  df-bl 17655  df-mopn 17656  df-fbas 17657  df-fg 17658  df-cnfld 17662  df-top 18344  df-bases 18346  df-topon 18347  df-topsp 18348  df-cld 18464  df-ntr 18465  df-cls 18466  df-nei 18543  df-lp 18581  df-perf 18582  df-cn 18672  df-cnp 18673  df-haus 18760  df-cmp 18831  df-tx 18976  df-hmeo 19169  df-fil 19260  df-fm 19352  df-flim 19353  df-flf 19354  df-xms 19736  df-ms 19737  df-tms 19738  df-cncf 20295  df-limc 21182  df-dv 21183  df-log 21892  df-cxp 21893  df-em 22270  df-cht 22318  df-vma 22319  df-chp 22320  df-ppi 22321  df-mu 22322

This theorem is referenced by: (None)